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Dr. Olga Ladyzhenskaya, 81, Mathematician

Dr. Olga Ladyzhenskaya, a mathematician whose work with differential equations contributed to advances in the study of fluid dynamics in areas like weather forecasting, oceanography, aerodynamics and cardiovascular science, died on Jan. 12 in St. Petersburg, Russia. She was 81.

The cause of death had not been determined, according to a spokeswoman for the Association for Women in Mathematics, in College Park, Md. Dr. Ladyzhenskaya was a member of the organization.

Her primary work was on calculations that were developed in the 19th century to explain the behavior of fluids and known as Navier-Stokes equations. As a researcher first at St. Petersburg University and later at the Steklov Institute of Mathematics, also in St. Petersburg, she worked through the solutions for the equations, which show how a number of variables relate in time and space.

Among other practical uses, the equations enable meteorologists to predict the movement of storm clouds.

In the 1960's, Dr. Ladyzhenskaya published her observations in a text that is still cited in the field. ''Ladyzhenskaya did not describe the basic equations, but she contributed significantly to their solutions,'' said Dr. Peter D. Lax of the Courant Institute of Mathematical Sciences at New York University. ''She was also always a rebel and treated as one by the Soviet government.''

Olga Aleksandrovna Ladyzhenskaya graduated from Moscow State University and received a doctorate from Leningrad State University before earning another doctorate from Moscow State in 1953. After teaching in the physics department at St. Petersburg University, she joined the Steklov Institute, which is affiliated with the Russian Academy of Sciences.

Dr. Marshall Slemrod, a mathematician with the University of Wisconsin, said Dr. Ladyzhenskaya had an American counterpart in John Nash, the Princeton mathematician and Nobel laureate whose life is depicted in the film ''A Beautiful Mind,'' and who also studied partial differential equations.

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''She was perhaps the premier worker on the Russian side,'' Dr. Slemrod said. ''If you believe your weather forecast, you have to solve the exact equations that she studied.''

Her later work involved the study of elliptical and parabolic equations that are used in probability theory.

Dr. Ladyzhenskaya's reputation as an independent spirit was furthered by her friendship with Aleksandr I. Solzhenitsyn, the author and dissident, and by reports that her father had been killed by Soviet officials, Dr. Lax said.

She was head of the Steklov Institute's laboratory of mathematical physics and was made a corresponding member of the Russian Academy of Sciences in 1981, before becoming a full member in 1990.

Earlier in her life, Dr. Ladyzhenskaya was briefly married. She has no immediate survivors.

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A version of this obituary; biography appears in print on January 25, 2004, on Page 1001038 of the National edition with the headline: Dr. Olga Ladyzhenskaya, 81, Mathematician. Order Reprints|Today's Paper|Subscribe